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<li><a class="reference internal" href="#">Gaussian Mixture Model Ellipsoids</a></li>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-mixture-plot-gmm-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="gaussian-mixture-model-ellipsoids">
<span id="sphx-glr-auto-examples-mixture-plot-gmm-py"></span><h1>Gaussian Mixture Model Ellipsoids<a class="headerlink" href="#gaussian-mixture-model-ellipsoids" title="Permalink to this headline">¶</a></h1>
<p>Plot the confidence ellipsoids of a mixture of two Gaussians
obtained with Expectation Maximisation (<code class="docutils literal notranslate"><span class="pre">GaussianMixture</span></code> class) and
Variational Inference (<code class="docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code> class models with
a Dirichlet process prior).</p>
<p>Both models have access to five components with which to fit the data. Note
that the Expectation Maximisation model will necessarily use all five
components while the Variational Inference model will effectively only use as
many as are needed for a good fit. Here we can see that the Expectation
Maximisation model splits some components arbitrarily, because it is trying to
fit too many components, while the Dirichlet Process model adapts it number of
state automatically.</p>
<p>This example doesn’t show it, as we’re in a low-dimensional space, but
another advantage of the Dirichlet process model is that it can fit
full covariance matrices effectively even when there are less examples
per cluster than there are dimensions in the data, due to
regularization properties of the inference algorithm.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">itertools</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>

<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">mixture</span>

<span class="n">color_iter</span> <span class="o">=</span> <span class="n">itertools</span><span class="o">.</span><span class="n">cycle</span><span class="p">([</span><span class="s1">&#39;navy&#39;</span><span class="p">,</span> <span class="s1">&#39;c&#39;</span><span class="p">,</span> <span class="s1">&#39;cornflowerblue&#39;</span><span class="p">,</span> <span class="s1">&#39;gold&#39;</span><span class="p">,</span>
                              <span class="s1">&#39;darkorange&#39;</span><span class="p">])</span>


<span class="k">def</span> <span class="nf">plot_results</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y_</span><span class="p">,</span> <span class="n">means</span><span class="p">,</span> <span class="n">covariances</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">title</span><span class="p">):</span>
    <span class="n">splot</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">index</span><span class="p">)</span>
    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">covar</span><span class="p">,</span> <span class="n">color</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span>
            <span class="n">means</span><span class="p">,</span> <span class="n">covariances</span><span class="p">,</span> <span class="n">color_iter</span><span class="p">)):</span>
        <span class="n">v</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">covar</span><span class="p">)</span>
        <span class="n">v</span> <span class="o">=</span> <span class="mf">2.</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">2.</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
        <span class="n">u</span> <span class="o">=</span> <span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
        <span class="c1"># as the DP will not use every component it has access to</span>
        <span class="c1"># unless it needs it, we shouldn&#39;t plot the redundant</span>
        <span class="c1"># components.</span>
        <span class="k">if</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">Y_</span> <span class="o">==</span> <span class="n">i</span><span class="p">):</span>
            <span class="k">continue</span>
        <span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">Y_</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">Y_</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="o">.</span><span class="mi">8</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>

        <span class="c1"># Plot an ellipse to show the Gaussian component</span>
        <span class="n">angle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
        <span class="n">angle</span> <span class="o">=</span> <span class="mf">180.</span> <span class="o">*</span> <span class="n">angle</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span>  <span class="c1"># convert to degrees</span>
        <span class="n">ell</span> <span class="o">=</span> <span class="n">mpl</span><span class="o">.</span><span class="n">patches</span><span class="o">.</span><span class="n">Ellipse</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mf">180.</span> <span class="o">+</span> <span class="n">angle</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
        <span class="n">ell</span><span class="o">.</span><span class="n">set_clip_box</span><span class="p">(</span><span class="n">splot</span><span class="o">.</span><span class="n">bbox</span><span class="p">)</span>
        <span class="n">ell</span><span class="o">.</span><span class="n">set_alpha</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
        <span class="n">splot</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">ell</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="o">-</span><span class="mf">9.</span><span class="p">,</span> <span class="mf">5.</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">6.</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(())</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>


<span class="c1"># Number of samples per component</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">500</span>

<span class="c1"># Generate random sample, two components</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">C</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.1</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.7</span><span class="p">,</span> <span class="o">.</span><span class="mi">4</span><span class="p">]])</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">C</span><span class="p">),</span>
          <span class="o">.</span><span class="mi">7</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">3</span><span class="p">])]</span>

<span class="c1"># Fit a Gaussian mixture with EM using five components</span>
<span class="n">gmm</span> <span class="o">=</span> <span class="n">mixture</span><span class="o">.</span><span class="n">GaussianMixture</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">covariance_type</span><span class="o">=</span><span class="s1">&#39;full&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">plot_results</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">gmm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">gmm</span><span class="o">.</span><span class="n">means_</span><span class="p">,</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span>
             <span class="s1">&#39;Gaussian Mixture&#39;</span><span class="p">)</span>

<span class="c1"># Fit a Dirichlet process Gaussian mixture using five components</span>
<span class="n">dpgmm</span> <span class="o">=</span> <span class="n">mixture</span><span class="o">.</span><span class="n">BayesianGaussianMixture</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
                                        <span class="n">covariance_type</span><span class="o">=</span><span class="s1">&#39;full&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">plot_results</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dpgmm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">dpgmm</span><span class="o">.</span><span class="n">means_</span><span class="p">,</span> <span class="n">dpgmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span>
             <span class="s1">&#39;Bayesian Gaussian Mixture with a Dirichlet process prior&#39;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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